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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.7.58
Chapter 2, Problem 2.7.58

Use the definitions given in Exercise 57 to prove the following infinite limits.


lim x→1^+ 1 /1 − x=−∞

Verified step by step guidance
1
Understand the problem: We need to prove that as x approaches 1 from the right (x → 1^+), the expression 1/(1-x) approaches negative infinity.
Consider the behavior of the denominator (1-x) as x approaches 1 from the right. Since x is slightly greater than 1, (1-x) becomes a small negative number.
Analyze the fraction 1/(1-x): As (1-x) becomes a small negative number, the fraction 1/(1-x) becomes a large negative number because dividing by a small negative number results in a large negative value.
Formalize the limit: For any large negative number M, we need to find a δ > 0 such that for all x satisfying 0 < x - 1 < δ, the inequality 1/(1-x) < M holds true.
Conclude the proof: By choosing δ = 1 - (1/M), we ensure that as x approaches 1 from the right, 1/(1-x) becomes less than any large negative number M, thus proving the limit is negative infinity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Infinite Limits

Infinite limits describe the behavior of a function as it approaches a certain point, where the function's value increases or decreases without bound. In this case, as x approaches 1 from the right (1+), the function 1/(1-x) tends toward negative infinity, indicating that the values of the function decrease indefinitely.
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One-Sided Limits

One-Sided Limits

One-sided limits focus on the behavior of a function as it approaches a specific point from one direction only. The notation lim x→1^+ indicates that we are considering values of x that are greater than 1, which is crucial for understanding how the function behaves as it nears the point of interest from the right side.
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One-Sided Limits

Understanding the Function's Behavior

To analyze the limit lim x→1^+ 1/(1-x), it is essential to understand how the denominator behaves as x approaches 1 from the right. As x gets closer to 1, 1-x approaches 0, causing the fraction to grow larger in magnitude and negative, leading to the conclusion that the limit is negative infinity.
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Graphs of Exponential Functions
Related Practice
Textbook Question

Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.


g(x)=∣x+4x2−4∣g\(\left\)(x\(\right\))=\(\left\[\vert\]\frac{x+4}{x^2-4}\[\right\]\vert\)

Textbook Question

Determine the following limits.

lim x→1 x^3 − 7x^2 + 12x / 4 − x

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Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

f(x)=sinxf\(\left\)(x\(\right\))=\(\sin\) x

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Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 x − 3 /|x − 3|

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Textbook Question

Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4

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Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→3 −5x / √4x − 3

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